Bayesian hierarchical models and prior elicitation for fitting psychometric functions

Abstract

Our previous articles demonstrated how to analyze psychophysical data from a group of participants using generalized linear mixed models (GLMM) and two-level methods. The aim of this article is to revisit hierarchical models in a Bayesian framework. Bayesian models have been previously discussed for the analysis of psychometric functions although this approach is still seldom applied. The main advantage of using Bayesian models is that if the prior is informative, the uncertainty of the parameters is reduced through the combination of prior knowledge and the experimental data. Here, we evaluate uncertainties between and within participants through posterior distributions. To demonstrate the Bayesian approach, we re-analyzed data from two of our previous studies on the tactile discrimination of speed. We considered different methods to include a priori knowledge in the prior distribution, not only from the literature but also from previous experiments. A special type of Bayesian model, the power prior distribution, allowed us to modulate the weight of the prior, constructed from a first set of data, and use it to fit a second one. Bayesian models estimated the probability distributions of the parameters of interest that convey information about the effects of the experimental variables, their uncertainty, and the reliability of individual participants. We implemented these models using the software Just Another Gibbs Sampler (JAGS) that we interfaced with R with the package rjags. The Bayesian hierarchical model will provide a promising and powerful method for the analysis of psychometric functions in psychophysical experiments.

Keywords: Bayesian model; PSE; generalized linear mixed models; psychometric functions; psychophysics.

Figure 1

Posterior estimates of parameters b^h (slope). Experiment in Section 3.1.

Figure 2

Posterior estimates of parameters PSE^h. Experiment in Section 3.1.

Figure 3

Posterior estimates of individual parameters of $ps{e}_i^h$. The (left) figure illustrated with red lines represents conditions with masking vibrations, while the (right) figure illustrated with blue lines represents conditions without masking vibrations. Experiment in Section 3.1.

Figure 4

Posterior estimates of individual parameters of ${\beta }_i^h$. The (left) figure illustrated with red lines represents conditions with masking vibrations while the (right) figure illustrated with blue lines represents conditions without masking vibrations. Experiment in Section 3.1.

Figure 5

Psychometric functions of individual participants from Experiment 1 in conditions without masking vibrations. The scatter plot shows the observed (dots) versus predicted responses (solid lines) with data from individual participants illustrated in each panel. Blue lines correspond to the prediction by GLMM, while red lines correspond to predictions by the Bayesian model. Experiment in Section 3.1.

Figure 6

Psychometric functions of individual participants from Experiment 1 in conditions with 32 Hz masking vibrations. The scatter plot shows the observed (dots) versus predicted responses (solid lines) with data from individual participants illustrated in each panel. Blue lines correspond to the prediction by GLMM, while red lines correspond to predictions by the Bayesian model. Experiment in Section 3.1.

Figure 7

Posterior distributions of parameters $b_k^h$ from the second stage of the hierarchical model. Experiment in Section 3.2.

Figure 8

Posterior distributions of parameters of the second stage of the hierarchical model $\text{PS}{\text{E}}_k^h$. Experiment in Section 3.2.

Figure 9

Posterior distributions of parameters of the first stage of the hierarchical model ${\beta }_i^h$, by group and masking condition. Experiment in Section 3.2.

Figure 10

Posterior distributions of parameters of the first stage of the hierarchical model $PS{E}_i^h$, by group and masking condition. Experiment in Section 3.2.

Figure 11

Posterior distributions of parameters PSE^h with different prior distributions for different values of a₀. The model with the informative prior (a₀ = 1.0) is illustrated in dark brown, the one with the power prior (α₀ = 0.7) in orange, and the one with the non-informative prior (α₀ = 0.0) in yellow. Experiment in Section 3.2.

Figure 12

Posterior distributions of parameters b^h with different prior distributions for different values of a₀. The model with the informative prior (a₀ = 1.0) is illustrated in dark brown, the one with the power prior (a₀ = 0.7) in orange, and the one with the non-informative prior (a₀ = 0.0) in yellow. Experiment in Section 3.2.